Activity Based Aggregate Job Costing Model for

International Journal of Industry and Sustainable Development (IJISD)

Volume 1, No. 2, PP. 1-19, July 2020, ISSN 2682-4000

Recommended by the IJISD Editor in Chief

http://ijisd.journals.ekb.eg 1

Activity Based Aggregate Job Costing Model for Reconfigurable Manufacturing Systems

Darwish Alami1 and Waguih ElMaraghy1, *

1 Industrial and Manufacturing Systems Engineering, University of Windsor,

401 Sunset Ave. Windsor, Ontario N9B 3P4, Canada

* Corresponding author: [email protected]

Abstract

Manufacturing continues to face escalated cost challenges as the global economy grows. In order to

gain competitive advantage among its rivals, manufacturing firms are in a constant strive to lower their

manufacturing costs compared to their competitors. This paper introduces a mathematical optimization model

based on Activity Based Costing (ABC) method for Reconfigurable Manufacturing Systems (RMS) taking into

consideration the bi- directional relationship between hourly rates and annual hours on each

machine/workcentre. The output from the model will be the optimum hourly rates, decision on which jobs to

accept or reject and decision on the financial feasibility of reconfiguration. Reconfiguration in this paper

describes both system-level reconfiguration (investing in additional machining equipment) and/or, machine-level

reconfiguration (extra module to an existing equipment). The model will be applied on a real life case study of a

global Original Equipment Manufacturer of Machinery. The novelty of the proposed model is the incorporation

of the bi-directional relationship between hourly rates and annual hours on each machine and provides a

managerial decision making tool in terms of investment level required to pursue new business, and gaining

competitive advantage over rivals.

Key words: Activity Based Costing, Reconfigurable Manufacturing Systems, Direct cost

1- Introduction

Manufacturing firms are striving nowadays to deliver a portfolio of products to its

customers with attainable prices to increase the market share of their products. Since the

product development process includes various numbers of interconnected activities, it is of a

great challenge to manufacturing firms to trace and allocate the different activities to cost

objects in order to specify the price of their products and services.

Cost accounting is the type of cost incurred after the product is manufactured. This

type of cost is prepared by accountants. There are several types of cost accounting. Process

costing method is employed when a standard product is being made which involves a number

of distinct processes performed in a definite sequence. Process costing is applied mainly in

continuous manufacturing (i.e. oil refinery). It is applied to manufacturing environment in

which similar products are produced. Job costing is concerned with finding the cost of each

individual job or contract. Batch costing is a form of job costing. Instead of costing each

component separately, each batch of components are taken together and treated as a job.

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Hybrid costing is a combination of the above.

In addition, there exist three main cost systems: (i) traditional costing system, (ii)

Activity Based Costing (ABC) and (iii) variable based costing [1, 2]. Monroy el al. [3]

presented the three different accounting systems’ approaches to manufacturing; (i) Activity

Based Costing, (ii) Time Driven Activity Based Costing and (iii) Lean Accounting.

As lean manufacturing implies identification of non-value added activities and reduce

them (if not eliminating them), lean accounting can also be defined as removing or

eliminating waste within the accounting process itself. There are three supporting key points

for applying lean accounting in lean organization: visual management, value stream

management and continuous improvement [4]. Throughput accounting cost includes direct

materials and direct labor which is used in Just In Time manufacturing environment.

Throughput is defined as the difference between revenue and the total variable expenses as

per [5]. The main difference between the traditional accounting and the throughput

accounting costing is the bottleneck operation or drum. For the throughput accounting

technique, it mandates optimizing the bottleneck operation only since any local optimization

of non-bottleneck operations will result in buffer accumulation. Unlike the traditional

accounting cost, which requires all firm operations to run as efficient as possible (as well as

non-bottleneck operations [5] . It has been reported by [6] that the usage of throughput

costing aided companies in reducing lead time, inventory and cycle time while increasing

productivity and quality. More on product costing methods is shown in Fig. 1.

Fig. 1. The different types of product costing [7]

This paper introduces a novel mathematical model to minimize the total cost incurred

in a job shop environment across production periods. The cost objects in this case refers to the

jobs being processed within the manufacturer’s floor (cost objects refers to an entity in which

cost measurement is desired such as products, services, activities and customers [8]). The

mathematical model takes into account the bi-directional relationship between hourly rates

and the total assigned hours as well as reconfiguration decision within machine and system

Non-factory costs (sales, admin, distribution) R&D, design, customer service, disposal

Fixed Factory Overhead

Variable Factory Overhead

Direct Material Direct Labour

Throughput costing

Variable costing

Full absorption costing

Life cycle costing and Activity based costing





level. Hence, this paper deals with job costing type of cost accounting due to its relevance

with job shop environment. In addition, the cost system used is ABC due to its accuracy in

allocating overhead costs to cost objects. In addition, workcentres and engineering

department capacities are taken into consideration as constraints. The mathematical model

developed for this paper is non-linear and hence, in order to convert it to a Mixed Linear

Integer Programming model, several linearization techniques will be applied.

This paper is organized as follows; section 2 is concerned with reviewing the previous

work and identifying the gap, section 3 provides a background to two main elements in this

paper which are the ABC method and manufacturing systems, section 4 is concerned with the

mathematical model development, section 5 presents a case study for applying the

mathematical model, section 6 lists the results and discussions from the case study and

finally, section 7 is the conclusion.

2- Literature Survey

Lin et al. [9] provided a model for integrating manufacturing and production system

performance cost. The integration between manufacturing and costing was achieved by

linking the Activity Based Costing (ABC) and Design for Manufacturing (DFM). Ramadan et

al. [10] proposed a real time manufacturing cost estimation method using RFID. It was

concluded that the proposed real time manufacturing tracking system is beneficiary in

identifying causes for redundant cost which can be an enabler for lean manufacturing.

Myrelid and Olhager [11] proposed a hybrid cost approach for mixed process environment

(job shop, flow shops and assembly lines) in order to establish a cost allocation for products

produced by manufacturer. Traditional, lean and throughput accounting approaches were used

and applied on three different products with varying complexity. Mathematical formulae were

formulated for each approach. They concluded that lean accounting cost model is to be

allocated to assembly lines, throughput accounting cost model to flow shops and traditional

accounting cost to job shop environment. Elsukova [12] illustrated the lean and throughput

cost accounting approach and proposed a framework for integrating both approaches. The

author concluded that the lean and throughput cost accounting approaches supplement one

another as the throughput cost accounting determines the improvement required for the flow

of material (restricted by bottlenecks) and lean cost accounting is mandatory to improve

production and reduce waste. Sajadfar and Ma [13] proposed a framework for cost estimation

for welded features using data mining and linear regression to come up with a feature cost

estimation. Jiang et al. [14] proposed a manufacturing cost model which takes into

consideration labor, material and overhead costs for mycelium-based bio-composite sandwich

structures. Duran and Afonso [15] proposed an Activity Based Costing and Life Cycle

Costing (LCC) model as a decision making tool for management of non-repairable spare

parts. Santana et al. [16] proposed a mathematical model incorporating Activity Based

Costing and Time Based Activity Based Costing for capacity management optimization. The

trade-off between capacity maximization and operational efficiency has been analyzed and

the authors suggested that capacity should be optimized rather than maximized since

maximizing capacity can lead to operational inefficiency. Tsai et al. [17] proposed a Green

Activity Based Costing (ABC) model applied within aluminum alloy wheel industry. The





model traces direct and indirect product costs to cost objects as well as allocates carbon tax to

cost objects. Tsai and Lai [18] proposed a mathematical programming model combining

green manufacturing technologies (i.e.), Activity Based Costing (ABC) and theory of

constraint in order to provide optimal production plans based on optimal profitable product

mix decision. The labour rates in the model have been considered as an input parameter to the

model and the bi-directional effect between labour rate and hours assigned to workcentres

was not considered.

From the conducted literature review, the bi-directional between direct hourly rates

and total hours assigned to departments/workcentres has not been discussed before. In

addition, this paper incorporates the machine/module reconfiguration cost, depending on

financial feasibility, if they will increase assigned hours in facility and hence reducing the

hourly rates.

3- Activity Based Costing

The activity based costing, originally introduced by [19, 20], works differently from

the traditional costing system. The main benefits of the ABC costing is the allocation of unit

cost of a product based on the capacity used for product or job. It starts by defining the

different activities involved in production (e.g. setup, machining….etc.), compute the cost for

each activity and then allocate each activity to its corresponding cost objects such as product

or job. This type of system works well for companies producing a wide scope of product

variants or jobs. The steps for ABC can be shown in Fig. 2. The main drawback of the ABC

method is its complexity in identifying the various activities which is time consuming and

requires high data processing costs.

For the ABC method, the hourly rate in specific period for a certain workcentre or

engineering activity is proposed as:

𝐻𝑜𝑢𝑟𝑙𝑦 𝑟𝑎𝑡𝑒 𝑓𝑜𝑟 𝑐𝑒𝑟𝑡𝑎𝑖𝑛 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝑜𝑟 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 (𝑡) =

= 𝐵𝑙𝑒𝑛𝑑𝑒𝑑 𝑐𝑜𝑠𝑡 +𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑎𝑠𝑠𝑒𝑡𝑠 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑒𝑑 𝑐𝑜𝑠𝑡𝐴𝐵𝐶 + 𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛/𝑦𝑟

𝑡𝑜𝑡𝑎𝑙 ℎ𝑜𝑢𝑟𝑠 𝑏𝑦 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝑜𝑟 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 (𝑡 − 1)

(1)

In Equation (1), the “General assets allocated cost ABC” is the portion of the general

assets allocated to a certain workcentre or activity. It is defined as the product of the total

general assets in a specific production period and the ratio between the hours assigned to a

workcentre/activity to the total hours by all workcentres/activities. As per Equation (1), it is

evident that hourly rate in a certain production period decrease as the total hours by machine

or activity increase in the previous production period.





Fig. 2. Activity Based Costing steps

4- Manufacturing Systems

Manufacturing systems have evolved along decades depending on customers’

requirement and needs in each era and through each industrial revolution. In industry 1.0,

steam power was the main enabler for industry 1.0 in an attempt to fulfil production volume

to the market. The production system type during industry 1.0 was called a craft production

system. With the introduction of electricity and mechanical devices, industry 2.0 took over in

which dedicated manufacturing lines were capable of satisfying production volume and

variety. Afterwards, with the advancements in micro processing and computers, industry 3.0

evolved with Flexible Manufacturing Systems as its main essence with focus on production

volume, variety and delivery time. Industry 4.0 is the new industrial revolution enabled by the

breakthrough in Information and Communication Technology (ICT), Internet of Things (IoT),

Cloud Computing and Cyber-Physical Systems [21]. Reconfigurable Manufacturing Systems

and Distributed Manufacturing Systems are the main production systems type in Industry 4.0.

Dedicated manufacturing lines (DML), or transfer lines [22], are based on inexpensive

fixed automation and produce a company's core products or parts at high volume. Each

dedicated line is typically designed to produce a single part (i.e., the line is rigid) at high

production rate achieved by the operation of several tools simultaneously in machining

stations (called "gang drilling"). When the product demand is high, the cost per part is

relatively low. DMLs are cost effective as long as demand exceeds supply and they can

operate at their full capacity. But with increasing pressure from global competition and over-

capacity built worldwide, there may be situations in which dedicated lines do not operate at

full capacity. Flexible manufacturing systems (FMS)[22, 23] can produce a variety of

products, with changeable volume and mix, on the same system. FMSs consist of expensive,





general-purpose computer numerically controlled (CNC) machines and other programmable

automation. Because of the single-tool operation of the CNC machines, the FMS throughput

is lower than that of DML. The cost per part relatively high due to the high cost of the general

purpose machine and low throughput. Reconfigurable Manufacturing Systems is type of

system characterized by rapid adjustability of functionality and capacity to meet changing

demand [23, 24]. The design of the reconfigurable manufacturing systems is intended for

part family, unlike DML and FMS which mainly focus on single part and general purpose

machine, respectively. There are several characteristics and enablers that qualify a system as

reconfigurable. These characteristics are [24]:

Modularity: modular system components to facilitate adjustment of the system

capacity and capability (adding/removing system components).

Integrability: all system components must be easily integrated through appropriate

interfaces.

Convertibility: quick changeover when changing between products (mixed model

production)

Diagnosability: quick identification of errors or malfunctions

Customization: match system capability and capacity to the product demand

Scalability: incremental increase or decrease in system capacity by adding or

removing system components easily.

Focused Flexibility Manufacturing Systems [25] is a hybrid type of manufacturing

systems in which Flexible Manufacturing Systems (FMS) exist with Dedicated

Manufacturing Lines and hence, flexibility is introduced not only through the individual

general purpose machines (e.g. CNC), but from the interaction between the two systems. Job

Shop [26] consists of a group of general purpose machines (e.g. CNC) together with often

dedicated equipment to mainly suit low volume production with high variety. Normally, there

is no specific type of flow in job shops due to its nature as a make-to-order type of facility

which depends on the customer’s orders (daily orders can vary from full size presses to small

size spare parts). This leads to complex scheduling and material handling within the shop.

Cellular Manufacturing Systems [27] is based on grouping of part families which are similar

in shape, material and manufacturing process and assign them to a group of machines known

as cells. A key enabler of cellular manufacturing is group technology. Group technology is a

concept in which relies on grouping parts sharing similar design, material and manufacturing

process into part families.

5- Mathematical Model Formulation

This section lists and illustrates the mathematical model implemented in this paper.

The IDEF0 model is shown in Fig. 3. The detailed description of the model, inputs and

outputs will be illustrated in the following subsections. The proposed model is non-linear. In

order to obtain the linear form and convert it to a Mixed Integer Linear Programming (MILP)

model, reader can refer to linearization methods in [28].





Fig. 3. IDEF0 model for the proposed mathematical model

The list of input parameters, decision variables, sets, constants objective function,

constraints are detailed in this section. The list of input parameters is:

𝑒𝑖,𝑜 = {1,0,

if workcentre o can operate on job i

otherwise

𝑓𝑚,𝑖,𝑜: Quoted/budget hours required for workcentre o to complete

job i when module m is added

𝑔𝑖,𝑜: Quoted/budget hours required for workcentre o to complete job i

𝑔𝑖,𝑝𝑁𝐸𝑊: Quoted/budget hours required for workcentre p to complete job i

ℎ𝑖,𝑗: Quoted/budget hours required for engineering dept. j to complete job i

𝑘𝑖,𝑜: Quoted/budget hours required to setup workcentre o for job i

𝑘𝑖,𝑝𝑁𝐸𝑊: Quoted/budget hours required to setup workcentre p for job i

𝑄𝑖,𝑡: Production demand/quantity of job i in production period t

𝑑𝑖,𝑡: Raw material/commercial items cost for job i in production period t

𝑠𝑖,𝑡: Selling price of job i in production period t

𝐶𝑜,𝑡𝑀 : Available capacity for workcentre o in production period t

𝐶𝑝,𝑡𝑀𝑁𝐸𝑊: Available capacity for workcentre p in production period t

𝐶𝑗,𝑡𝐸 : Available capacity for engineering department j in production period t

𝐷𝐸𝑃𝑜𝐸𝑊𝑆: Depreciation of existing workcentre o

𝐷𝐸𝑃𝑝𝑁𝑊𝑆: Depreciation of new workcentre p

Minimize the total job

costs (Labour and

material cost) taking

reconfiguration into

consideration

Accepted orders

New workcentres hourly rates

Existing workcentres hourly rates

Available orders

Workcentres Capacity

Activity Based

Costing

MILP

Workcentre/functional module reconfiguration decision

Cloud Computing

Total hours for each order on available m/c’s

Workcentre/functional module reconfiguration cost

General assets and equipment depreciation

Total hours for each order on new m/c’s





𝐷𝐸𝑃𝑗𝐸𝑁𝐺 : Depreciation of equipment in engineering department j

𝐺𝐴𝑜,𝑡𝐸𝑊𝑆: General assets allocation cost to existing workcentre o in period t

𝐺𝐴𝑝,𝑡𝑁𝑊𝑆: General assets allocation cost to new workcentre p in period t

𝐺𝐴𝑗,𝑡𝐸𝑁𝐺: General assets allocation cost to department j in period t

𝑐𝑚𝑀𝑂𝐷: purchase cost of functional module m

𝑐𝑝𝑀𝐶: purchase cost of new workcentre p

The decision variables are:

𝑎𝑗,𝑡: Hourly rate for engineering department j in production period t

𝑏𝑜,𝑡: Hourly rate for workcentre o in production period t

𝑏𝑝,𝑡𝑁𝐸𝑊: Hourly rate for new workcentre p in production period t

𝐵𝑢𝑦𝑜,𝑚𝑡𝑀𝑂𝐷 = {

1,0,

if module m is bought for workcentre o in production period t

otherwise

𝐵𝑢𝑦𝑝,𝑡𝑀𝐶 = {

1,0,

if new workcentre p is bought in production period t

otherwise

𝑀𝑂𝐷𝑜,𝑚,𝑡 = {1,0,

if module m is required in workcentre o in production period t

otherwise

𝑀𝐶𝑝,𝑡 = {1,0,

if workcentre o is required in production period t

otherwise

𝑥𝑖,𝑡 = {1,0,

if job i is chosen in production period t

otherwise

The objective function is concerned with maximizing the profit in which it is required

to minimize the difference between total cost and total selling price. Several assumptions are

considered while formulating the objective function. For example, raw materials are

purchased for each job and therefore, no carrying or holding cost is considered in this paper.

The objective function is written as shown in Equation (2).

The mathematical model only considers mechanical and electrical engineering as well

as manufacturing as direct labour cost. Factory overhead and indirect costs are allocated to

the direct hourly rates. In Equation (2), the first term is the raw material/commercial items

cost. The second term is the total engineering cost. In ABC method terms, second term is the

product level. The third term is the total production cost. In ABC method terms, the third term

is the unit level. The fourth term is the total setup cost. In ABC method terms, the fourth term

is the batch level. The fifth and sixth terms are the buying costs of functional modules and

adding new workcentres, respectively. Finally, the seventh term is the selling cost. In the





fourth term, the symbol ε is a small number (e.g. 0.0005) to prevent an infinite value for the

fourth term in case where Qi,t is equal to zero.

Min Z

∑ ∑ 𝑑𝑖,𝑡𝑄𝑖,𝑡𝑥𝑖,𝑡

𝑇

𝑡=1+ ∑ ∑ ∑ ℎ𝑖,𝑗𝑎𝑗,𝑡𝑥𝑖,𝑡

𝑇

𝑡=1

𝐽

𝑗=1

𝐼

𝑖=1

𝐼

𝑖=1

+ ∑ ∑ ∑ 𝑄𝑖,𝑡𝑔𝑖,𝑜𝑏𝑜,𝑡𝑥𝑖,𝑡

𝑇

𝑡=1

𝑂

𝑜=1

𝐼

𝑖=1+ ∑ ∑ ∑

(𝑘𝑖,𝑜𝑏𝑜,𝑡𝑥𝑖,𝑡)

𝑄𝑖,𝑡 + 𝜀

𝑇

𝑡=1

𝑂

𝑜=1

𝐼

𝑖=1

+ ∑ ∑ ∑ 𝑐𝑚𝑀𝑂𝐷𝐵𝑢𝑦𝑜,𝑚𝑡

𝑀𝑂𝐷𝑇

𝑡=1

𝑀

𝑚=1

𝑂

𝑜=1+ ∑ ∑ 𝑐𝑝

𝑀𝐶𝑇

𝑡=1𝐵𝑢𝑦𝑝,𝑡

𝑀𝐶𝑃

𝑝=1

− ∑ ∑ 𝑠𝑖,𝑡𝑄𝑖,𝑡𝑥𝑖,𝑡

𝑇

𝑡=1

𝐼

𝑖=1

(2)

The constraints for the proposed model are:

𝑎𝑗,𝑡 = 𝑎0 +𝐺𝐴𝑗,𝑡

𝐸𝑁𝐺 + 𝐷𝐸𝑃𝑗𝐸𝑁𝐺

∑ 𝑥𝑖,𝑡0ℎ𝑖,𝑗

𝐼𝑖=1

, ∀𝑗 = 1,2. . 𝐽, 𝑡 = 1,2. . 𝑇, 𝑡0 = 𝑡 − 1

(3)

𝑏𝑝,𝑡𝑁𝐸𝑊 = 𝑏0

𝑁𝐸𝑊 +𝐺𝐴𝑝,𝑡

𝑁𝑊𝑆 + 𝐷𝐸𝑃𝑝𝑁𝑊𝑆

∑ 𝑥𝑖,𝑡0𝑄𝑖,𝑡0

𝑔𝑖,𝑝𝑁𝐸𝑊𝐼

𝑖=1

,

∀ 𝑝 = 1,2. . 𝑃, 𝑡 = 1,2. . 𝑇, 𝑡0 = 𝑡 − 1 (4)

𝑏𝑜,𝑡 = 𝑏0 +𝐺𝐴𝑜,𝑡

𝐸𝑊𝑆 + 𝐷𝐸𝑃𝑜𝐸𝑊𝑆

∑ 𝑥𝑖,𝑡0𝑄𝑖,𝑡0

(𝑔𝑖,𝑜𝐼𝑖=1 + ∑ 𝑓𝑚,𝑖,𝑜𝑀𝑂𝐷𝑜,𝑚,𝑡0

𝑀𝑚=1 )

,

∀𝑜 = 1,2. . 𝑂, 𝑡 = 1,2. . 𝑇, 𝑡0 = 𝑡 − 1 (5)

∑ (𝑔𝑖,𝑝𝑁𝐸𝑊𝑄𝑖,𝑡𝑥𝑖,𝑡 + 𝑘𝑖,𝑝

𝑁𝐸𝑊𝑥𝑖,𝑡) ≤ 𝐶𝑝,𝑡𝑀𝑁𝐸𝑊

𝐼

𝑖=1, ∀ 𝑝 = 1,2. . 𝑃, 𝑡 = 1,2. . 𝑇

(6)

∑ (𝑔𝑖,𝑜𝑄𝑖,𝑡𝑥𝑖,𝑡 + ∑ 𝑓𝑚,𝑖,𝑜𝑀𝑂𝐷𝑜,𝑚,𝑡

𝑀

𝑚=1𝑄𝑖,𝑡𝑥𝑖,𝑡 + 𝑘𝑖,𝑜𝑥𝑖,𝑡) ≤ 𝐶𝑡,𝑜

𝑀𝐼

𝑖=1

∀𝑜 = 1,2. . 𝑂, 𝑡 = 1,2. . 𝑇 (7)

∑ ℎ𝑖,𝑗𝑥𝑖,𝑡 ≤ 𝐶𝑗,𝑡𝐸

𝐼

𝑖=1, ∀ 𝑗 = 1,2. . 𝐽, 𝑡 = 1,2. . 𝑇

(8)

𝑀𝑂𝐷𝑜,𝑚,𝑡 ≤ ∑ 𝑓𝑚,𝑖,𝑜𝑥𝑖,𝑡(1 − 𝑒𝑖,𝑜) ≤ 𝑀𝑂𝐷𝑜,𝑚,𝑡𝐵𝑖𝑔𝑀𝐼

𝑖=1,

∀ 𝑜 = 1,2. . 𝑂, 𝑡 = 1,2. . 𝑇, 𝑚 = 1,2. . 𝑀 (9)





𝐵𝑢𝑦𝑜,𝑚,𝑡𝑀𝑂𝐷𝐵𝑖𝑔𝑀 − 𝐵𝑖𝑔𝑀 + 1 ≤ 𝑀𝑂𝐷𝑜,𝑚,𝑡 − 𝑀𝑂𝐷𝑜,𝑚,𝑡0

≤ (1 − 𝜀)(1 − 𝐵𝑢𝑦𝑜,𝑚,𝑡𝑀𝑂𝐷) + 𝐵𝑢𝑦𝑜,𝑚,𝑡

𝑀𝑂𝐷 ,

∀ 𝑜 = 1,2. . 𝑂, 𝑚 = 1,2. . 𝑀, 𝑡 = 1,2. . 𝑇, 𝑡0 = 𝑡 − 1 (10)

𝐵𝑢𝑦𝑝,𝑡𝑀𝐶𝐵𝑖𝑔𝑀 − 𝐵𝑖𝑔𝑀 + 1 ≤ 𝑀𝐶𝑝,𝑡 − 𝑀𝐶𝑝,𝑡0

≤ (1 − 𝜀)(1 − 𝐵𝑢𝑦𝑝,𝑡𝑀𝐶) + 𝐵𝑢𝑦𝑝,𝑡

𝑀𝐶 ,

∀𝑝 = 1,2. . 𝑃, 𝑡 = 1,2. . 𝑇, 𝑡0 = 𝑡 − 1 (11)

∑ 𝑥𝑖,𝑡 ≥ 1𝐼

𝑖=1, ∀𝑖 = 1,2. . 𝐼, 𝑡 = 1,2. . 𝑇

(12)

Equation (3) is an equality constraint. It is used to represent the bi-directional between

the hourly rates for department j in production period t and the hours assigned to department j

in the previous production period t0. Similarly, Equation (4) represents the bi-directional

relationship between hourly rates for new workcentre p in production period t and the hours

assigned to new workcentre p in the previous production period t0. Equation (5) represents the

bi-directional relationship between hourly rates for workcentre o in production period t and

the hours assigned to workcentre o in the previous production period t0.

Equation (6) represents the capacity of the new workcentre p in production period t in

hours. Equation (7) represents the capacity of the existing workcentre o in production period

t. The left hand side of Equation (7) is composed of three terms. The first term (gi,oQi,txi,t)

represents the total hours required for job i on existing workcentre o. The second term

(∑m=1,2,..Mfm,i,oMODo,m,tQi,txi,t) denotes the hours from job i added to workcentre o when

adding additional functional module m in production period t. The third term (ki,oxi,t) is the

setup hours required by job i on workcentre o.

Equation (8) is the available capacity in department j during production period t.

Equation (9) represents the condition in which a functional module m is required by

workcentre o in production period t. If functional module m when added to workcentre o can

machine job i (i.e. fm,i,o>0) but workcentre o without the additional functional module m

cannot machine job i (i.e.1-ei,o=1), therefore, functional module m is required by workcentre

o (i.e. MODm,o,t=1). Equation (10) represents the condition in which purchasing of additional

functional module m in production period t for workcentre o takes place (Buymodo,m,t). If in

two subsequent periods t and t+1, functional module m is required by workcentre o (i.e.

MODo,m,t+1=1 and MODo,m,t=1), then there is no purchasing of the additional functional

module taking place in production period t+1 (i.e. Buymodo,m,t+1=0). However, if functional

module m is not required by workcentre o in production period t (i.e. MODo,m,t=0), but

functional module m is required by workcentre o in production period t+1 (i.e.

MODo,m,t+1=1), therefore, purchasing a new module is required in production period t+1 (i.e.

Buymodo,m,t+1=1).d Equation (11) represents the condition in which a new workcentre p is to

be purchased in production period t. This constraint can be illustrated similar to constraint

Equation (9).

Finally, Equation (12) represents the condition in which at least one job is selected in

each production period. It is evident that several non-linear terms exist in the constraints

equations. For example, the second term on the left hand side of Equation (7)

(∑m=1,2,..Mfm,i,oMODo,m,tQi,txi,t) is composed of two binary variables multiplied together

(MODo,m,t and xi,t) which requires obtaining such variables in their linear form in order to use





MILP. Reader may refer to [28] for further readings on linearization techniques.

6- Industrial Case Study

The case study discussed in this paper is adopted from a local machine shop and is

part of a multinational machinery builder company situated in Europe. The company shop

area is around 5,000 m2 with various departments such welding, fabrication, machining and

assembly. The inputs to the model are shown in Table 1 to Table 8. The name of each

workcentre is described as a symbol WS as actual names of workcentres were not allowed to

be disclosed. As per Equation (1), the hourly rate for a certain workcentre at a specific

production period is calculated based on the assigned hours for the workcentre in the previous

period. Hence, the hourly rate for workcentres and departments for production period 1 are

considered constant and the values are shown in Table 1 and Table 2. The blended costs for

existing workcentres, new workcentres and engineering department b0, b0NEW and a0,

respectively, are taken as $60/hr. The number production periods considered are six

production periods in which each period is considered a quarter of a year.

Table 1. General assets allocated cost to each existing workcentre and depreciation cost/yr for each

existing workcentre in ($)

index Hourly

rate at t=1 Existing

Workcentre

General assets

allocated Cost

($)

Depreciation/yr

($)

o=1 $62 WS-01 $20,141.36 $-

o=2 $62 WS-02 $59,057.32 $-

o=3 $65 WS-03 $59,057.32 $-

o=4 $100 WS-04 $14,061.27 $29,180.00

o=5 $65 WS-05 $15,186.17 $110,020.00

o=6 $85 WS-06 $14,061.27 $29,180.00

o=7 $85 WS-07 $8,436.76 $3,440.00

o=8 $125 WS-08 $7,030.63 $73,630.00

o=9 $85 WS-09 $15,186.17 $187,950.00

o=10 $62 WS-10 $20,141.36 $2,710.00

o=11 $62 WS-11 $59,057.32 $-

o=12 $62 WS-12 $59,057.32 $-

o=13 $62 WS-13 $59,057.32 $-

o=14 $62 WS-14 $59,057.32 $-

o=15 $62 WS-15 $59,057.32 $-

o=16 $62 WS-16 $20,141.36 $-

o=17 $62 WS-17 $20,141.36 $-

o=18 $62 WS-18 $20,141.36 $-

Table 2. General assets allocated cost to each new workcentre and depreciation cost/yr for each new

workcentre in ($)

Workcentre

Index

Hourly

rate at t=1 Workcentre

General assets

allocated Cost

($)

Depreciation/yr

($)

p=1 $62 WS-01 $20,141.36 $29,180

p=2 $62 WS-02 $59,057.32 $29,180

p=3 $65 WS-03 $59,057.32 $3,000





Table 3. Hours required by engineering departments 1 and 2 to complete job i

Job

i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9 i=10

Dept. j=1 610 690 770 920 560 520 170 760 510 110

j=2 500 300 810 180 320 280 840 960 390 820

Table 4. Available jobs required machining hours on each new workcentre, selling price in $ and raw

material cost in $

Job

index

new workcentres available Selling price

in ($)

Raw material

cost in ($) p=1 p=2 p=3

i=1 $1,347,660 $36,866

400 1755 785

i=2 $1,254,225 $97,531

657.5 1040 592.5

i=3 $1,954,685 $21,615

400 100 500

i=4 $1,373,780 $99,247

400 890 500

i=5 $1,303,530 $96,111

400 297.5 250

i=6 $471,850 $35,654

600 935 1520

i=7 $1,961,505 $32,793

400 200 5539

i=8 $717,165 $69,500

147.5 330 55

i=9 $645,540 $99,464

195 340 50

i=10 $1,555,845 $31,337

135 370 15

Table 5. Sample machining and setup hours (between brackets) required for job i on existing workcentre o

Available workcentre

Job

index o=1 o=2 o=10 o=11 o=15 o=16 o=18

i=1 82.048(2) 1755(1) 1240(1) 670(1) 80(1) 267.5(1) 50(1)

i=2 657.5(2) 1040(2) 0(0) 760.5(2) 756(2) 677.5(1) 0(0)

i=3 60(2) 100(2) 100(2) 150(1) 50(2) 0(0) 200(2)

i=4 145(1) 890(1) 200(1) 150(1) 300(1) 105(1) 40(2)

i=5 115(2) 297.5(2) 200(1) 192.5(2) 300(2) 100(2) 110(2)

i=6 30(1) 935(2) 90(1) 1107.5(2) 82.5(1) 80(2) 880(1)

i=7 0(0) 0(0) 0(0) 0(0) 0(0) 0(0) 0(0)

i=8 147.5(1) 330(2) 185(2) 80(1) 142.5(1) 70(1) 50(1)

i=9 195(2) 340(2) 230(2) 202.5(1) 167.5(1) 75(1) 10(1)

i=10 135(2) 370(1) 250(1) 100(2) 167.5(1) 70(1) 20(2)





Table 6. Available capacity for existing workcentres and engineering departments in production periods 1

up to 6 in hours

Existing workcentres

Departments 1,3,12

13,14 4,6 11,17

15,16 2 9,10 5 7 8 18 1 2

t=1 to

t=6 9600 7680 4800 2880 2160 3840 1920 2000 2000

The general assets allocated to workcentres and activities are shown in Table 1 and

Table 2. Though the general assets allocated to workcentre/activities must be calculated as per

illustration in section 3, it is taken directly from company’s records as constant to avoid

further linearization of terms in the model.

Table 7. Available capacity for new workcentres available for purchasing in production periods 1 up to 6

in hours

Production

periods

New workcentres

available

p=1 p=2 p=3

t=1 to t=6 9600 7680 9600

Table 8. Machining hours required by each available job when functional module m=1 is added to

existing workcentres

Job Available workcentres

index o=4 o=5 o=6 o=7 o=8 o=9

i=1 795 40 600 0 0 760

i=2 1267 1900 3365 895 0 600

i=3 0 0 70 300 0 0

i=4 200 105 0 0 0 1850

i=5 100 100 0 0 0 1500

i=6 140 505 1670 472.5 0 720

i=7 0 0 0 0 0 0

i=8 680 750 50 100 150 400

i=9 40 750 250 0 180 400

i=10 110 750 250 200 190 400

7- Results and Discussions

The objective function is concerned with maximizing the profit in which it is required

to minimize the difference between total cost and total selling price. Several assumptions are

considered while formulating the objective function. The Mixed Integer Linear Programming

(MILP) model is written using AMPL (http://ampl.com/) and solved using Gurobi in Neos

[29-31].

The results from the model are presented in Fig. 4 to Fig. 8 and Table 9. Fig. 4 shows

the hourly rates for the engineering departments during the 6 production periods. The first

period is taken as the blended cost as illustrated in Equation (1). The hourly rate increases to

the maximum at period 5 for the two departments. Since each hourly rate is calculated based

on the total hours in the previous period, therefore, the hourly rates for period 5 are calculated


http://ampl.com/




based on the hours from period 4. From Table 9 and Table 3, the total engineering hours for

departments 1 and 2 are 1640 and 1500 hours, respectively compared to 1970 and 1940 hours

for periods 1, 2, 3 and 5 for departments 1 and 2, respectively. Therefore, in order to reduce

the hourly rate further, the manufacturing firm is required to accept more jobs from customers

in order to reduce the hourly rates.

Table 9. Jobs accepted in each production period

Jobs

i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9 i=10

t=1 1 1 0 0 1 0 0 0 0 1

Periods t=2 1 1 0 0 1 0 0 0 0 1

t=3 1 1 0 0 1 0 0 0 0 1

t=4 1 0 0 1 0 0 0 0 0 1

t=5 1 1 0 0 1 0 0 0 0 1

t=6 0 0 1 1 0 0 1 0 0 0

Fig. 4. Hourly rates for engineering departments 1 and 2 during the 6 production periods

Fig. 7 presents the results of the machining hourly rates variation with the different

production periods. The machining hourly rates peaks at production period 5 due to the

reduced assigned total machining hours in production period 4 which is equal to 5250 hours

compared to 6697.5 hours for production periods 1, 2, 3 and 5 and 8929 hours for production

period 6. Fig. 8 presents the reconfiguration level decision on the machine and system level.

In production period 1, the mathematical model suggests purchasing three new workcentres

Fig. 5 and Fig. 6 shows the hourly rates for workcentres WS-01 to WS-09 and WS-10

to WS-18, respectively. From Table 9 and Table 5, the total machining hours in production

period 4 sums up to 11249.55 hours compared to 18178.55 hours for production periods 1, 2,

59

60

61

62

63

64

65

1 2 3 4 5 6

Ho

url

y R

ate

Production periods

Engg. Dept. 1 Engg. Dept. 2





3 and 5. Therefore, the hourly rates for production period 5 peaks to the maximum value as

per Equation (1). For hourly rates of WS-04 and WS-07 in Fig. 5, the hourly rates are

decreased in production period 2 since the total machining hours in production period 1 is at

its maximum of 181,718.55 hours.

Fig. 5. Hourly rates for existing workcentres WS-01 to WS-09 in the 6 production periods

The reason for this decision is to increase the available machining capacity within the

facility, and hence, more jobs can be accepted. The model also suggests adding functional

module 1 to the existing workcentres WS-04, WS-05, WS-06, WS-07 and WS-09. The reason

for this decision is to extend the functionality of the existing workcentres in order to accept

jobs containing new features that cannot be machined by the existing system’s capability.

To this extent, the more jobs being accepted, the more reduction in job cost will be

encountered as a result of reducing the hourly machining rates. It is worth noting that

reducing hourly rates and hence, job cost, will put the manufacturing company at a

competitive edge among its rivals due to following the cost leadership business strategy and

accordingly, higher probability to get more jobs from customers.

0

50

100

150

200

250

1 2 3 4 5 6

Ho

url

y R

ate

Production periods

WS-01 WS-02 WS-03 WS-04 WS-05

WS-06 WS-07 WS-08 WS-09





Fig. 6. Hourly rates for existing workcentres WS-10 to WS-18 in the 6 production periods

Fig. 7. Hourly rates for the new workcentres in the 6 production periods

0

50

100

150

200

250

300

1 2 3 4 5 6

Ho

url

y R

ate

Production periods

WS-10 WS-11 WS-12 WS-13 WS-14

WS-15 WS-16 WS-17 WS-18

0

20

40

60

80

100

120

1 2 3 4 5 6

Ho

url

y R

ate

Production periods

WS-01 (new) WS-02 (new) WS-03 (new)





Fig. 8. Workcentres/modules reconfiguration decision

8- Conclusion

This paper introduces a novel mathematical model to minimize the total cost incurred

in a Reconfigurable Manufacturing System environment across production periods. The cost

objects, in this case, refer to the jobs being processed within the manufacturer’s floor. The

objective function developed is to minimize the total manufacturing cost and increase profit

through a proposed ABC model. In addition, the mathematical model takes into consideration

the bi-directional relationship between the number of hours assigned to

workcentres/departments and the hourly rates. The main outputs from the mathematical

model are the newly calculated hourly rates for the different workcentres/departments, the

jobs mix decision as well as functional modules and workcentres reconfiguration decisions.

The proposed mathematical model is applied to a case study taken from a local heavy

machinery builder machine shop.

The significance of this paper is not restricted to cost analysis, but also to provide

managers in manufacturing facilities with the required decision-making tools to decide on

orders to accept or refuse as well as investing in additional production equipment. In

addition, this paper will assist manufacturing companies to achieve a competitive edge among

rivals through reduction of hourly rates within their facility. Additional benefits and

significance are (1) providing manufacturing companies a method to quantify the decision-

making process for right-sizing their manufacturing space (2) ability to justify growing a

scalable system using costing (not customer demand) (3) expanding market share and (4)

reducing operational cost and allowing companies a numerical model to justify scaling the

manufacturing system.

WS-01

WS-02

WS-03

WS-04

WS-05

WS-06

WS-07

WS-08

WS-09

WS-10

WS-11

WS-12

WS-13

WS-14

WS-15

WS-16

WS-17

WS-18

Existing Workcentres

WS-01

WS-02

WS-03 Module

1





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